Question: Compute
\[\sum_{n = 1}^\infty \frac{2n - 1}{n(n + 1)(n + 2)}.\]
Answer: First, we split $\frac{2n - 1}{n(n + 1)(n + 2)}$ into partial fractions by writing
\[\frac{2n - 1}{n(n + 1)(n + 2)} = \frac{A}{n} + \frac{B}{n + 1} + \frac{C}{n + 2}.\]Then $2n - 1 = A(n + 1)(n + 2) + Bn(n + 2) + Cn(n + 1).$

Setting $n = 0,$ we get $-1 = 2A,$ so $A = -\frac{1}{2}.$

Setting $n = -1,$ we get $-3 = -B,$ so $B = 3.$

Setting $n = -2,$ we get $2C = -5,$ so $C = -\frac{5}{2}.$  Thus,
\[\frac{2n - 1}{n(n + 1)(n + 2)} = -\frac{1/2}{n} + \frac{3}{n + 1} - \frac{5/2}{n + 2}.\]Therefore,
\begin{align*}
\sum_{n = 1}^\infty \frac{2n - 1}{n(n + 1)(n + 2)} &= \left( -\frac{1/2}{1} + \frac{3}{2} - \frac{5/2}{3} \right) + \left( -\frac{1/2}{2} + \frac{3}{3} - \frac{5/2}{4} \right) \\
&\quad + \left( -\frac{1/2}{3} + \frac{3}{4} - \frac{5/2}{5} \right) + \left( -\frac{1/2}{4} + \frac{3}{5} - \frac{5/2}{6} \right) + \dotsb \\
&= -\frac{1}{2} + \frac{5/2}{2} = \boxed{\frac{3}{4}}.
\end{align*}